
so that \(X(t), t\in [0,\infty )\)  lies in the positive half line \((0, \infty )\)  almost surely. On the other hand, if \(0<\alpha <2\) , \({\rm P}(T_0^{x,\alpha }<\infty )=1\)  and \(X\)  is recurrent, see [1]}. Hence, to make \(H_{\alpha }\) (t) well-defined, (REF ) is defined up to time \(T_0:=\min _{1\le k \le N-1}T_0^{x_k,\alpha _k}\)  when there exists \(k\)  such that \(0<\alpha _k<2\) . Note that \(H_{\alpha }(t)\)  can be seen as a time evolution of G\(\beta \) E because if we take \(\alpha =((N-1)\beta , (N-2)\beta , \dots , 2\beta , \beta )\)  for some \(\beta >0\) , by (REF ) we have \(\frac{1}{\sqrt{\beta }}(H_{\alpha }(1)-H_{\alpha }(0))=H_{\beta }^{tri}\)  in distribution.
